Multidimensional Hopfield Networks for clustering

We present the Multidimensional Hopfield Network (DHN), a natural generalisation of the Hopfield Network. In our theoretical investigations we focus on DHNs with a certain activation function and provide energy functions for them. We conclude that these DHNs are convergent in finite time, and are equivalent to greedy methods that aim to find graph clusterings of locally minimal cuts. We also show that the general framework of DHNs encapsulates several previously known algorithms used for generating graph embeddings and clusterings. Namely, the Cleora graph embedding algorithm, the Louvain method, and the Newman's method can be cast as DHNs with appropriate activation function and update rule. Motivated by these findings we provide a generalisation of Newman's method to the multidimensional case.